arXiv subjects

Hyunjin In

Publications and source records attributed to Hyunjin In.

4 recordsLinked to original sources

arXiv2026

A Lagrangian Approach to Uniqueness for Active Scalars in Generalized Mean Oscillation Spaces

We develop a unified Lagrangian framework for the uniqueness of weak solutions to a broad class of two-dimensional inviscid active scalar equations in generalized mean oscillation spaces. Under common structural assumptions on the Biot--Savart kernels, we identify mean oscillation spaces that ensure an Osgood condition for the velocity, and establish uniqueness in these classes. For the two-dimensional Euler equations, we establish uniqueness for vorticities whose mean oscillation grows mildly at small scales, extending beyond $\mathrm{BMO}$. For the surface quasi-geostrophic (SQG) and incompressible porous media (IPM) equations, we obtain uniqueness in spaces logarithmically rougher than $C^{0,1}$. We also obtain what are (to our knowledge) the strongest known uniqueness results for the $α$-SQG equation, $0<α<1$, in spaces logarithmically rougher than critical Hölder spaces $C^α$. The framework also applies to various logarithmically perturbed models, yielding uniqueness in the corresponding sharp regularity classes. The proof combines a sharp regularity estimate for the velocity with a stability estimate for integral operators under measure-preserving rearrangements. This provides a single uniqueness argument for the entire family under common structural assumptions on the kernels.

math.AP↗

arXiv2025

Asymptotic stability of the $2D$ temperature-dependent tropical climate model with the sharp decay rates

We investigate the asymptotic stability of a tropical climate model posed on $\bR^2$, with temperature-dependent diffusion in the barotropic mode $u$ and linear damping in the first baroclinic mode $v$. We consider two distinct cases for the barotropic component: one with linear damping and one without. For both cases, we prove the small data global existence of smooth solutions. Furthermore, we establish sharp temporal decay estimates for solutions in arbitrary Sobolev norms $H^m (\bR^2)$, $m \ge 0$.

math.AP↗

arXiv2024

New Liouville-type theorem for the stationary tropical climate model

We study the Liouville-type theorem for smooth solutions to the steady 3D tropical climate model. We prove the Liouville-type theorem if a smooth solution satisfies a certain growth condition in terms of $L^p$-norm on annuli, which improves the previous results, Theorem 1.1 (Math. Methods Appl. Sci. 44, 2021) by Ding and Wu, and Theorem 1.1 and Theorem 1.2 (Appl. Math. Lett. 138, 2023) by Yuan and Wang.

math.AP↗

arXiv2023

An improved Liouville-type theorem for the stationary tropical climate model

In this paper, we study the Liouville-type property for smooth solutions to the steady 3D tropical climate model. We prove that if a smooth solution $(u,v,θ)$ satisfies $u \in L^3 (\mathbb{R}^3)$, $v \in L^2 (\mathbb{R}^3)$, and $\nabla θ\in L^2 (\mathbb{R}^3)$, then $u=v=0$ and $θ$ is constant, which improves the previous result, Theorem 1.3 (Math. Methods Appl. Sci. 44, 2021) by Ding and Wu.

math.AP↗